Data.Colour.Matrix:inverse from colour-2.3.3, B

Average Error: 7.4 → 1.7

Time: 2.4min

Precision: binary64

Cost: 1800

Math

\[\frac{x \cdot y - z \cdot t}{a} \]

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := y \cdot \frac{x}{a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t \ne 0:\\ \;\;\;\;\frac{z}{\frac{a}{-t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \end{array} + t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t + t_2\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))) (t_2 (* y (/ x a))))
   (if (<= t_1 -2e+120)
     (+ (if (!= t 0.0) (/ z (/ a (- t))) (/ (* t (- z)) a)) t_2)
     (if (<= t_1 5e+170) (/ t_1 a) (+ (* (/ (- z) a) t) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = y * (x / a);
	double tmp_1;
	if (t_1 <= -2e+120) {
		double tmp_2;
		if (t != 0.0) {
			tmp_2 = z / (a / -t);
		} else {
			tmp_2 = (t * -z) / a;
		}
		tmp_1 = tmp_2 + t_2;
	} else if (t_1 <= 5e+170) {
		tmp_1 = t_1 / a;
	} else {
		tmp_1 = ((-z / a) * t) + t_2;
	}
	return tmp_1;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - (z * t)) / a
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_1 = (x * y) - (z * t)
    t_2 = y * (x / a)
    if (t_1 <= (-2d+120)) then
        if (t /= 0.0d0) then
            tmp_2 = z / (a / -t)
        else
            tmp_2 = (t * -z) / a
        end if
        tmp_1 = tmp_2 + t_2
    else if (t_1 <= 5d+170) then
        tmp_1 = t_1 / a
    else
        tmp_1 = ((-z / a) * t) + t_2
    end if
    code = tmp_1
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = y * (x / a);
	double tmp_1;
	if (t_1 <= -2e+120) {
		double tmp_2;
		if (t != 0.0) {
			tmp_2 = z / (a / -t);
		} else {
			tmp_2 = (t * -z) / a;
		}
		tmp_1 = tmp_2 + t_2;
	} else if (t_1 <= 5e+170) {
		tmp_1 = t_1 / a;
	} else {
		tmp_1 = ((-z / a) * t) + t_2;
	}
	return tmp_1;
}

Python

def code(x, y, z, t, a):
	return ((x * y) - (z * t)) / a

↓

def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	t_2 = y * (x / a)
	tmp_1 = 0
	if t_1 <= -2e+120:
		tmp_2 = 0
		if t != 0.0:
			tmp_2 = z / (a / -t)
		else:
			tmp_2 = (t * -z) / a
		tmp_1 = tmp_2 + t_2
	elif t_1 <= 5e+170:
		tmp_1 = t_1 / a
	else:
		tmp_1 = ((-z / a) * t) + t_2
	return tmp_1

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	t_2 = Float64(y * Float64(x / a))
	tmp_1 = 0.0
	if (t_1 <= -2e+120)
		tmp_2 = 0.0
		if (t != 0.0)
			tmp_2 = Float64(z / Float64(a / Float64(-t)));
		else
			tmp_2 = Float64(Float64(t * Float64(-z)) / a);
		end
		tmp_1 = Float64(tmp_2 + t_2);
	elseif (t_1 <= 5e+170)
		tmp_1 = Float64(t_1 / a);
	else
		tmp_1 = Float64(Float64(Float64(Float64(-z) / a) * t) + t_2);
	end
	return tmp_1
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - (z * t)) / a;
end

↓

function tmp_4 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	t_2 = y * (x / a);
	tmp_2 = 0.0;
	if (t_1 <= -2e+120)
		tmp_3 = 0.0;
		if (t ~= 0.0)
			tmp_3 = z / (a / -t);
		else
			tmp_3 = (t * -z) / a;
		end
		tmp_2 = tmp_3 + t_2;
	elseif (t_1 <= 5e+170)
		tmp_2 = t_1 / a;
	else
		tmp_2 = ((-z / a) * t) + t_2;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+120], N[(If[Unequal[t, 0.0], N[(z / N[(a / (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(t * (-z)), $MachinePrecision] / a), $MachinePrecision]] + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 5e+170], N[(t$95$1 / a), $MachinePrecision], N[(N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Target

Original	7.4
Target	5.8
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -2e120

   1. Initial program 17.6

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Applied egg-rr 43.6

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{x \cdot y - z \cdot t}{a}\right)}^{3}}} \]

   3. Applied egg-rr 11.0

      \[\leadsto \color{blue}{\frac{z \cdot t}{-a} - \left(-y\right) \cdot \frac{x}{a}} \]

   4. Simplified 11.0

      \[\leadsto \color{blue}{\frac{z \cdot t}{-a} + y \cdot \frac{x}{a}} \]
      Proof

   5. Applied egg-rr 3.8

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;t \ne 0:\\ \;\;\;\;\frac{z}{\frac{a}{-t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ } \end{array}} + y \cdot \frac{x}{a} \]

   if -2e120 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.99999999999999977e170

   1. Initial program 1.0

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 4.99999999999999977e170 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 22.3

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Applied egg-rr 44.3

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{x \cdot y - z \cdot t}{a}\right)}^{3}}} \]

   3. Applied egg-rr 13.5

      \[\leadsto \color{blue}{\frac{z \cdot t}{-a} - \left(-y\right) \cdot \frac{x}{a}} \]

   4. Simplified 13.5

      \[\leadsto \color{blue}{\frac{z \cdot t}{-a} + y \cdot \frac{x}{a}} \]
      Proof

   5. Applied egg-rr 2.5

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot t} + y \cdot \frac{x}{a} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	1.4
Cost	1800

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{-t}{a} \cdot z + y \cdot \frac{x}{a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	1.1
Cost	1800

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{-z}{a} \cdot t + y \cdot \frac{x}{a}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+170}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	4.1
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]

Alternative 4
Error	20.4
Cost	1424

\[\begin{array}{l} t_1 := \frac{-t \cdot z}{a}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	18.0
Cost	1232

\[\begin{array}{l} t_1 := \frac{y \cdot x}{a}\\ \mathbf{if}\;x \cdot y \leq -20000:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \mathbf{elif}\;x \cdot y \leq 10^{-70}:\\ \;\;\;\;\frac{-t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \ne 0:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	31.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]

Alternative 7
Error	32.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-227}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y\\ \end{array} \]

Alternative 8
Error	32.5
Cost	320

\[\frac{x}{a} \cdot y \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))